22 GRAPHICAL METHODS IN THE 



But we may also estimate the work and heat of any path by means 

 of an integration extending over the elements of an area, viz : by the 

 formulae of page 7, 



r 



In regard to the limits of integration in these formulae, we see that for 

 the work of any path which is not a circuit, the bounding line is com- 

 posed of the path, the line of no pressure and two vertical lines, and 

 for the heat of the path, the bounding line is composed of the path, 

 the line of absolute cold and two horizontal lines. 



As the sign of y, as well as that of 8 A, will be indeterminate until 

 we decide in which direction an area must be circumscribed in order 

 to be considered positive, we will call an area positive which is cir- 

 cumscribed in the direction in which the hands of a watch move. 

 This choice, with the positions of the axes of volume and entropy 

 which we have supposed, will make the value of y in most cases posi- 

 tive, as we shall see hereafter. 



The value of y, in a diagram drawn according to this method, will 

 depend upon the properties of the body for which the diagram is 



drawn. M this respect, this method 

 differs from all the others which have 

 been discussed in detail in this article. 

 It is easy to find an expression for y 

 depending simply upon the variations of 

 N _ N the energy, by comparing the area and 



_ I the work or heat of an infinitely small 



N * N circuit in the form of a rectangle having 



its sides parallel to the two axes. 



Let N 1 N 2 N 3 N 4 (fig. 8) be such a circuit, 

 and let it be described in the order of 

 v the numerals, so that the area is positive. 

 Also let e v e 2 > e 3 , e 4 represent the energy 



at the four corners. The work done in the four sides in order com- 

 mencing at Np will be e 1 e 2 , 0, e 3 e 4 , 0. The total work, therefore, 

 for the rectangular circuit is 



Now as the rectangle is infinitely small, if we call its sides dv and dq, 

 the above expression will be equivalent to 



d z e 



-j 5- dv dn. 

 dvdrj 



